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← Older revision		Revision as of 00:40, 16 April 2013
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	~~Hi there! :~~) ~~My name~~ is ~~Hyman~~, I'm a ~~student studying Anthropology~~ and ~~Sociology from Pessac~~, ~~France~~.<br><br>~~Here~~ is ~~my webpage~~; [~~http~~://~~www~~.~~cologames~~.~~com/profile/624597/ScGame~~.~~html backup plugin~~]		In [[statistics]], in the analysis of two-way [[randomized block design]]s where the response variable can take only two possible outcomes (coded as 0 and 1), '''Cochran's Q test''' is a [[non-parametric]] [[statistical test]] to verify if ''k'' treatments have identical effects.<ref>{{cite book
			\|title = Practical Nonparametric Statistics
			\|author = Conover, William Jay
			\|ISBN = 9780471160687
			\|edition = Third Edition
			\|publisher = Wiley, New York, NY USA
			\|year = 1999
			\|pages = 388–395
			}}</ref><ref>National Institute of Standards and Technology. [http://www.itl.nist.gov/div898/software/dataplot/refman1/auxillar/cochran.htm Cochran Test]</ref> It is named for [[William Gemmell Cochran]]. Cochran's Q test should not be confused with [[Cochran's C test]], which is a variance outlier test.

			==Background==
			Cochran's Q test assumes that there are ''k'' > 2 experimental treatments and that the observations are arranged in ''b'' [[blocking (statistics)\|blocks]]; that is,

			{\| class="wikitable"
			\|-
			!
			! Treatment 1
			! Treatment 2
			! <math>\cdots</math>
			! Treatment ''k''
			\|-
			\| Block 1
			\| ''X''<sub>11</sub>
			\| ''X''<sub>12</sub>
			\| <math>\cdots</math>
			\| ''X''<sub>1''k''</sub>
			\|-
			\| Block 2
			\| ''X''<sub>21</sub>
			\| ''X''<sub>22</sub>
			\| <math>\cdots</math>
			\| ''X''<sub>2''k''</sub>
			\|-
			\| Block 3
			\| ''X''<sub>31</sub>
			\| ''X''<sub>32</sub>
			\| <math>\cdots</math>
			\| ''X''<sub>3''k''</sub>
			\|-
			\| <center><math>\vdots</math></center>
			\| <center><math>\vdots</math></center>
			\| <center><math>\vdots</math></center>
			\| <center><math>\ddots</math></center>
			\| <center><math>\vdots</math></center>
			\|-
			\| Block ''b''
			\| ''X''<sub>''b''1</sub>
			\| ''X''<sub>''b''2</sub>
			\| <math>\cdots</math>
			\| ''X''<sub>''b''k</sub>
			\|}

			==Description==
			Cochran's Q test is
			:H<sub>0</sub>: The treatments are equally effective.
			:H<sub>a</sub>: There is a difference in effectiveness among treatments.

			The Cochran's Q test statistic is
			:<math>
			T = k\left(k-1\right)\frac{\sum\limits_{j=1}^k \left(X_{\bullet j} - \frac{N}{k}\right)^2}{\sum\limits_{i=1}^b X_{i\bullet}\left(k-X_{i\bullet}\right)}
			</math>
			where
			:''k'' is the number of treatments
			:''X<sub>• j</sub>'' is the column total for the ''j''<sup>th</sup> treatment
			:''b'' is the number of blocks
			:''X<sub>i •</sub>'' is the row total for the ''i''<sup>th</sup> block
			:''N'' is the grand total

			==Critical region==
			For [[significance level]] α, the critical region is
			:<math>
			T > \Chi^2_{1-\alpha,k-1}
			</math>
			where Χ<sup>2</sup><sub>1 − α,k − 1</sub> is the (1 − α)-[[quantile]] of the [[chi-squared distribution]] with ''k'' − 1 degrees of freedom. The null hypothesis is rejected if the test statistic is in the critical region. If the Cochran test rejects the null hypothesis of equally effective treatments, pairwise [[multiple comparisons]] can be made by applying Cochran's Q test on the two treatments of interest.

			==Assumptions==
			Cochran's Q test is based on the following assumptions:
			#A large sample approximation; in particular, it assumes that ''b'' is "large".
			#The blocks were randomly selected from the population of all possible blocks.
			#The outcomes of the treatments can be coded as binary responses (i.e., a "0" or "1") in a way that is common to all treatments within each block.

			==Related tests==
			* When using this kind of design for a response that is not binary but rather ordinal or continuous, one instead uses the [[Friedman test]] or [[Durbin test]]s.
			* The case where there are exactly two treatments is equivalent to [[McNemar's test]], which is itself equivalent to a two-tailed [[sign test]].

			==References==
			{{Reflist}}

			{{NIST-PD}}

			[[Category:Statistical tests]]
			[[Category:Non-parametric statistics]]

99.231.50.3: typo (Modeh -> Model)

2008-12-10T03:11:36Z

typo (Modeh -> Model)

New page

Hi there! :) My name is Hyman, I'm a student studying Anthropology and Sociology from Pessac, France.<br><br>Here is my webpage; [http://www.cologames.com/profile/624597/ScGame.html backup plugin]

Equisatisfiability - Revision history

en>EmausBot: Bot: Migrating 1 interwiki links, now provided by Wikidata on d:Q5384599

99.231.50.3: typo (Modeh -> Model)